ODE/IM correspondence and Bethe ansatz for affine Toda field equations

We study the linear problem associated with modified affine Toda field equation for the Langlands dual $\hat{\mathfrak{g}}^\vee$, where $\hat{\mathfrak{g}}$ is an untwisted affine Lie algebra. The connection coefficients for the asymptotic solutions of the linear problem are found to correspond to the $Q$-functions for $\mathfrak{g}$-type quantum integrable models. The $\psi$-system for the solutions associated with the fundamental representations of $\mathfrak{g}$ leads to Bethe ansatz equations associated with the affine Lie algebra $\hat{\mathfrak{g}}$. We also study the $A^{(2)}_{2r}$ affine Toda field equation in massless limit in detail and find its Bethe ansatz equations as well as T-Q relations.


Introduction
The ODE/IM correspondence was proposed by Dorey and Tateo in [1] where they demonstrated an interesting relationship between a Schrödinger-type ordinary differential equation with anharmonic potential and the conformal limit of a certain two-dimensional quantum integrable model. It was shown that functional relations satisfied by the Stokes multipliers and spectral determinants of this ODE agree with those of the Q-operator and transfer matrix vacuum eigenvalues for an A 1 type quantum integrable system in the conformal field theory limit (see also [2]). The case where the Schrödinger differential equation is modified with an additional angular momentum potential was studied in [3].
This correspondence is now just a single example of the growing number of links between classical and quantum integrable models.
The generalization of this massless ODE/IM correspondence to simple Lie algebra A r was carried out in [4,5]. The case of other simple Lie algebras was studied in [6], where it was necessary to consider in general pseudo-differential equations. The work of [7] showed that the same results could be obtained by using a first order formulation that did not require introduction of a formal anti-derivative. Lukyanov and Zamolodchikov [8] studied the ODE/IM correspondence for the massive sine(h)-Gordon model and found that spectral determinants of a modified form of the classical sinh-Gordon model coincide with the Q-functions of the quantum sine-Gordon model, the affine Toda field theory for algebra A (1) 1 . This was generalized to a relation between the classical Tzitzéica-Bullough-Dodd equation (A (2) 2 algebra) and the quantum Izergin-Korepin model in [9], and was studied for type A (1) r affine Toda theories in [10,11]. In these works it was shown that connection coefficients for subdominant solutions to the linear problem associated with the affine Toda field equation correspond to the vacuum eigenvalues of Q-operators for g-type quantum integrable models. The work of [11] looked at ABCDG-type affine Lie algebras and found that the (pseudo-)ordinary differential equation associated withĝ ∨ affine Toda field equation was the same as that of [6] for simple Lie algebra g after taking the conformal limit.
While the work of [8,9] used a functional relation on the subdominant solution to the linear problem to obtain Bethe ansatz equations satisfied by the Q-function, the connection to the previously studied ψ-systems was not manifest. The ψ-system, a set of functional relations among uniquely defined solutions ψ (a) to a (pseudo-)ODE for a = 1, . . . , rank(g), was found in [6] (see also [7]). These ψ-systems are similar to the Plücker type relations, and using these relations they were able to derive the Bethe ansatz equations satisfied by the Q-functions which corresponded to the Q-function of a conformal vertex model associated to g.
In this paper we investigate the ψ-system of [6,7] and show how it also holds in the massive case for subdominant solutions to the linear problem associated to a modified affine Toda field equation for affine Lie algebraĝ ∨ , whereĝ is an untwisted affine algebra.
The case of A (2) 2r is unique in that it is non-simply laced yet its Langlands dual is equal to itself. Furthermore, the correspondence in [11] links massive theories associated to the Langlands dual affine algebraĝ ∨ to conformal quantum theories associated with g in the massless limit, so it is interesting to understand A (2) 2r which does not fit into this scheme in more detail. To investigate the meaning in this case we also propose a new ψ-system for A (2) 2r and give evidence for it by studying the spectral determinant of the ordinary differential equation associated with the linear problem and find its T -Q relations and the Bethe ansatz equations satisfied by Q. The case of untwisted non-simply laced affine Lie algebras remains elusive at the moment.
The flow of this paper is as follows. In section 2 we introduce the modified form of the classical affine Toda field equation used in this paper and its linear form. This section's main purpose is to introduce some special solutions to the linear problem determined by their asymptotic behavior near the irregular singularity at z = ∞ and the regular singularity at z = 0. Section 3 introduces the ψ-system functional relations satisfied by uniquely determined subdominant solutions to the linear problem Ψ (a) . These massive ψ-systems serve as the fulcrum of this work, linking the classical affine Toda differential equations with Q-functions corresponding to some massive quantum integrable model.
Finally section 4 uses the special solutions of section 2 and the functional relations of section 3 to give relations satisfied by the connection coefficients Q that are the same as Bethe ansatz equations for associated quantum integrable models.

Affine Toda field equations
In this section we will first summarize the Lie algebra conventions used in this paper.
We then introduce the modified affine Toda field equation, including its linear form, and study special solutions defined by their asymptotic behaviors.

Lie algebra preliminaries
A rank r Lie algebra g has generators in {E α , H i } where α ∈ ∆ (the set of roots) and i = 1, . . . , r. The commutation relations satisfied by these generators are [12] [ /α 2 is the coroot of α and N α,β are structure constants. Lie algebra g has fundamental weights ω a and simple roots α a where a = 1, . . . , r and α ∨ a · ω b = δ a,b . The Cartan matrix is defined to be A ab = α a · α ∨ b . We normalize the roots so that the long root has length 2.
Letĝ denote the affine Lie algebra of g. Its extended Dynkin diagram is obtained from that of g by adding the root α 0 = −θ, where θ is the highest root. The (dual) Coxeter labels n a (n ∨ a ) are integers satisfying 0 = r a=0 n a α a = r a=0 n ∨ a α ∨ a and n ∨ 0 = 1. The (dual) Coxeter number h (h ∨ ) is the sum of the (dual) Coxeter labels, and the (co)Weyl vector ρ (ρ ∨ ) is the sum of the (co)fundamental weights.ĝ ∨ denotes the Langlands dual ofĝ, whose simple roots are α ∨ a . The simply-laced affine Lie algebras A r , and E (1) r are self-dual, whereas the non simply-laced cases obey (B

Modified affine Toda field equation
First we will define the two-dimensional affine Toda field equation associated withĝ. The theory is defined on the complex plane using coordinates where ρ and θ are polar coordinates. The equation of motion for the two-dimensional modified affine Toda equation studied here is 1 The conformal factor p(z) in this equation is chosen to have the form (see [8,9]) Equation (2.5) can be written as a zero curvature condition, dA + A ∧ A = 0, where the one form A = A dz +Ā dz is This zero curvature condition can equivalently be written as a first order linear problem defined on some finite dimensional g-module, Such connections can be changed through an arbitrary gauge transformation of the form This leaves the zero curvature condition and linear problem unchanged, and will be used to put the connection into various convenient forms.

Asymptotic behavior
Now we will look at the asymptotic behavior of solutions to the modified affine Toda field equation and its linear problem.
First, following [8,9,10,11] we consider a special family of solutions to the equation of motion (2.5) φ(ρ, θ) with the following properties: (i) Consistent with the choice of p(z) in (2.6), φ(ρ, θ) should have periodicity: The field φ(ρ, θ) is real-valued for real ρ and θ (i.e. whenz is identified as the complex conjugate of z), and finite everywhere except at the apex ρ = 0.
The periodicity condition naturally leads one to define the following transformation under which both the equation of motion and linear problem are unchanged for integer k, (2.14) Functions that are rotated by this transformation are said to be k-Symanzik rotated, and will often be denoted with a subscript as follows, The linear problem also has another symmetry, Π : This symmetry follows naturally by noticing that underΠ, E α i transforms as For the following we will consider the linear problem (2.9) in g-module V (a) where the representation of this module has highest weight ω a and dimension α>0 (ωa+ρ)·α ρ·α [12] where ρ is the Weyl vector, half the sum of the positive roots. The vector space V (a) has a basis e In this work, we will be interested in the unique solution Ψ (a) in module V (a) that is subdominant, that is, the solution that decays fastest along the positive real axis. To find this subdominant solution it is useful to take a gauge transformation (2.10) that puts either the holomorphic or anti-holomorphic connection into a nice form with no exponentials where U is respectively (2.18) In the large z limit, φ(z,z) ∼ M ρ ∨ β log(zz) and p(z) ∼ z hM , and the connections becomẽ Now the subdominant solution is found to be, through consideration of the holomorphic and anti-holomorphic linear problems separately and then shifting back to the ± are the eigenvalues of Λ ± with the largest real part and its eigenvector in module V (a) . This eigenvalue is distinct, and furthermore since the representations can be chosen such that E ⊤ α = E −α , we have Λ − = (Λ + ) ⊤ and the two eigenvalues and eigenvectors are the same.
Finally, after setting the Ψ (a) from (2.21) and (2.22) to be equal, f and g are fixed within a constant giving ApplyingΩ k to this for any real number k gives the k-Symanzik rotated solution Note that aΠ transformation applied to Ψ (a) gives the same large-ρ behavior as Ψ k is the subdominant solution in the Stokes sector . (see [8,9]). By considering the holomorphic and anti-holomorphic linear problem it can be shown that such a solution X where the overall constant's dependence on λ was fixed by requiring that this solution is invariant underΩ k . Note however that the Ψ (a) solutions do not display this invariance form a basis of solutions to the linear problem, the subdominant solution Ψ (a) can be expanded as These coefficients Q We conjecture that for affine Toda field equations with algebraĝ ∨ these Q-functions will correspond to the vacuum eigenvalues of Q-operators for some massive integrable quantum field theory associated withĝ. We will give evidence for this correspondence by showing that in the conformal limit these connection coefficients Q will satisfy Bethe ansatz equations associated to vertex models with Langlands dual Lie algebra symmetry.

ψ-system
The ψ-system [6] is a set of Plücker type relations satisfied by auxiliary functions that are constructed from the subdominant solution to a (pseudo-)ODE. The ψ-system was proved for A-type simple Lie algebras and was conjectured for all other simple Lie algebras. In [7], the ψ-system for classical Lie algebras was derived by studying the first order system equivalent to the (pseudo-)ODE of [6] and embeddings of g-modules.
We will study the ψ-systems in the context of modified affine Toda field equations with algebraĝ ∨ and show that the same system of functional relations holds for the massive case. In particular it will be shown that the unique subdominant solutions Ψ (a) to the linear problem in g-module V (a) satisfy the same ψ-system relations of [7] forĝ ∨ when g is a classical Lie algebra, and [6] when g is an exceptional Lie algebra. We also find a new ψ-system for A (2) 2r affine Toda theories. Let us consider an embedding of modules as explained in [7] (see also [13]). In the r there is an embedding ι which acts as As consistency expects the highest weight of the left and right side modules are the same, ω a−1 + ω a+1 . Next, the incidence matrix B ab is related to the Cartan matrix as r the eigenvalues are such that the ψ-system with the same large ρ behavior on both sides is found to be ι Ψ where it is natural to define Ψ (0) and Ψ (r+1) to be 1. Here ι is the above embedding of modules.
The ψ-system can be written in a general form for any simply laced case as ι Ψ where the matrix B ab is defined in (3.3). When considering non-simply laced cases, there are difficulties that arise forĝ = B (1) 4 , and G 2 with deriving a Bethe ansatz equation that has only simple poles, so we will not consider these untwisted non-simply laced cases here.

Twisted cases
The ψ-systems for twisted cases can be found by computing the eigenvalue with largest real part of Λ + and are, 1/4 = Ψ (2) ⊗ Ψ

A
2r -type case This case does not fall under theĝ ∨ → g identification since there is no simple Lie algebra X r such that (X 2r . Nevertheless, a study of the eigenvalues of Λ + for this case show that the ψ-system that Ψ (a) satsifies is 2 ι Ψ When r = 1 this is the same functional relation as equation (4.77) in [9] for the case of the Tzitzéica-Bullough-Dodd model.

Bethe ansatz equations
Using the above ψ-systems, it is now possible to derive functional relations for the Qfunctions defined in equation (2.27) that will correspond to Bethe ansatz equations. We will verify that for modified affine Toda field equation with algebraĝ ∨ , when taking the conformal limit the Q-functions satisfy Bethe ansatz equations associated with g found in the context of the massless ODE/IM correspondence [6].
The conformal limit for modified affine Toda field equation with algebraĝ discussed in section 2 is reached using the following definitions, where ψ (a) and χ The definition of twisting used is from [14], where the role of α and α ∨ are swapped compared with [11]; here α 2 0 = 1 2 , α 2 i = 1, and α 2 r = 2, while n ∨ 0 = 1 and n ∨ i = 2.
1, . . . , dim V (a) ) can be determined through the linear problem and X (a) i , and is where ω ≡ e 2πi/h(M +1) and h is the Coxeter number for affine Lie algebraĝ. Under a k-Symanzik rotation in the conformal limit Q These functional relations are exactly the same as the Bethe ansatz equations for A r -type conformal vertex models.
This method applied to the other algebras give Bethe ansatz equations 2r−1 : 6 : 4 : Each of these Bethe ansatz equations agree with those reported in [6] (see also [15]) under the identificationĝ ∨ → g.
For the case of A (2) 2r , the same procedure gives Bethe ansatz equations (4.13) Note that for A 2 , which corresponds to the Tzitzéica-Bullough-Dodd model discussed in [9], α r = ω r − ω r−1 and (4.13) reduces to their equation (4.85). Since this case does not fall under the identificationĝ ∨ → g, it is important to verify these equations. To this end in appendix A we derived the T -Q relations that give rise to Bethe ansatz equations (4.13) starting from an analysis of the ODE itself and not using the ψ-system (see [16] for the A (2) 2 case). Furthermore, in appendix B we also show how these Bethe ansatz equations can be found in the work of [17] which looked at Bethe ansatz equations associated with twisted quantum affine Lie algebras.

Discussion
In this paper we studied a classical affine Toda field theory for affine Lie algebraĝ ∨ that is modified by a conformal transformation. Writing this modified affine Toda field equation in the linear form (d + A)Ψ = 0 translates the problem into a holomorphic and antiholomorphic first order matrix ordinary differential equation. Studying the asymptotic behavior of solutions Ψ to this linear problem, a unique subdominant solution Ψ (a) is found depending on the module V (a) in which the vector Ψ lives. These subdominant solutions Ψ (a) were then found to obey a set of functional relations, the massive ψ-system (see [6,7] for massless case). By expanding Ψ (a) in the basis of solutions X in this expansion. Substituting this expansion then into the ψ-system in the conformal limit gives a set of functional relations on the Q-functions that is of the same form as Bethe ansatz equations associated with a g-type conformal quantum vertex model. This was carried out for modified affine Toda field equations with algebraĝ ∨ where g is a simple Lie algebra and the resulting Bethe ansatz system matched those of [6,7] The presence of the Langlands dual affine algebra hints that the ODE/IM correspondence here could be a manifestation of Langlands duality [18].
This identification under the conformal limit gives important evidence in support of our conjecture that the proposed ψ-systems hold for massive systems and that the ODE/IM correspondence links the classical modified affine Toda equations to a massive quantum integrable model. Furthermore, previous work on the massive ODE/IM correspondence in this context on the modified sinh-Gordon equation [8] and A (1) r -type Toda theories [10] are in agreement with this work. A Also, the massive ODE/IM correspondence was recently studied in the case of the classical modified sinh-Gordon equation for a choice of p(z) defined on the 3-punctured Riemann sphere, and was found to correspond to the quantum Fateev model [19]. A generalization to affine Lie superalgebras [20] would also be interesting to study to explore the integrable structure of superstring theory in AdS space-time.

Note added:
During the preparation of this paper, we became aware of [21] where the conformal limit has been also studied for simply-laced cases.
A T -Q relations for A (2) 2r In this appendix we will derive the Bethe ansatz equations (4.13) for A (2) 2r starting from the ODE satisified by the top component of Ψ (1) in the conformal limit. This was done for the A (2) 2 case in [16]. For this discussion the ψ-system will not be used explicitly, but for reference we write down the ψ-system here where in the conformal limit it reduces to Wronskian relations on the top component of each vector Ψ (a) , In the conformal limit the top component of Ψ (1) , ψ (1) , satisfies the ODE (see [11]) This equation has a subdominant solution ψ with asymptotic behavior A Symanzik rotation of ψ(x, E, g) is defined to be ψ k (x, E, g) = ω −kr ψ(ω k x, ω hM k E, g) , ω = e 2πi/h(M +1) . This implies that the solutions {ψ k , ψ k+1 , . . . , ψ k+2r } are linearly independent. We will also make use of the notation and define the auxiliary functions which will make up the ψ-system To show that the above functions (A.8) asymptotically satisfy ψ-system (A.1), first note that the above ψ k asymptotic functions are exactly what one would get in the case of 2r . The work of [6,7] then gives a ψ system for auxiliary functions ψ su(h) . The twisting of A (1) 2r to A (2) 2r implies that we expect ψ (a) = ψ (2r+1−a) . Using the trigonometric relations one can show indeed that Notice that the coefficient in the exponential here is exactly µ (a) /µ (1) , as required. This demonstrates that after making the identification ψ (r+1) ∼ ψ (r) the ψ-system of A (1) 2r reduces to (3.12). In the case of A 2r one cannot truly identify ψ (r+1) ∼ ψ (r) , but for A (2) 2r in addition to the large x behavior the small x behavior is also in agreement, Now, using ψ-system (A.1) the Bethe ansatz equations (4.13) can be proven to hold through the T -Q relations we will now derive. Since {ψ k , ψ k+1 , . . . , ψ k+2r } form a basis of solutions, we can expand ψ as Then, using the notation W k,k+1,...,k+a−1 and determinant relations in [4] gives where T (1) (E) ≡ C (1) (ω −hM E) and the Coxeter number h is 2r + 1 in this case.
We will also expand ψ (a) in terms of solutions defined by the small x behavior as i . (A.14) After considering just the most divergent first term (i = 1) in this expansion, we can then make the identification (Q In this appendix we show how the Bethe ansatz equations for A (2) 2r (4.13) are in agreement with [17], which looked at Bethe ansatz equations for twisted quantum affine algebras (see also [15]).
The Bethe ansatz equation associated to a solvable vertex model associated with the twisted quantum affine algbra U q (A (2) 2r ) [17] is . (B.8) In the A 2r case, for these Bethe ansatz equations to agree with (4.13), simultaneously replace E (a) j → −E (a) j for odd a and setθ = π − θ and take N and N (1) to be even.
The identification then holds for Q